By the independence of \(h^0\)  and \(\mathfrak {h}\)  we have from [1]} that
\(\mathbf {E}\!\left[ \mu _{h^w}(A_0) \, | \, \mathfrak {h}\right] = \int _{A_0} e^{\gamma \mathfrak {h}(z)} r_{{S}}(z)^{\gamma ^2/2} dz\) 

where \(r_{S}(z)\)  is the conformal radius of \({S}\)  as seen from \(z\) .
Let \(\widetilde{\mathcal {Q}}\)  be a Whitney square decomposition of \(A_0\) . Fix \(\widehat{\delta } > 0\)  and let \(\mathcal {Q}\)  be formed by subdividing each \(Q \in \widetilde{\mathcal {Q}}\)  into a finite number of subsquares (depending only on \(\widehat{\delta }\) ) of equal size so that if \(Q \in \mathcal {Q}\)  then \(\mathrm {diam}(Q) / \mathrm {dist}(Q,\partial {S}) \le \widehat{\delta }\) . Fix some \(p \in (0,1)\)  to be determined later. By the concavity of the function \(x \mapsto x^p\)  for \(p \in (0,1)\)  we have that
\(\left( \int _{A_0} e^{\gamma \mathfrak {h}(z)} r_{{S}}(z)^{\gamma ^2/2} dz \right)^p \lesssim \sum _{Q \in \mathcal {Q}} e^{\gamma p \sup _{z \in Q} |\mathfrak {h}(z)|} \mathrm {dist}( Q, \partial {S})^{p \gamma ^2/2} | Q |^p,\) 

where \(|Q|\)  is the area of \(Q\) . Taking expectations, we have that
\(\mathbf {E}\!\left[ \mathbf {E}\!\left[ \mu _{h^w}(A_0) \, | \, \mathfrak {h}\right]^p \right] \lesssim \sum _{Q \in \mathcal {Q}} \mathbf {E}\!\left[e^{\gamma p \sup _{z \in Q} |\mathfrak {h}(z)|} \right] \mathrm {dist}( Q, \partial {S})^{p \gamma ^2/2} | Q |^p.\) 

Next, note that \(h^w\)  differs from a free boundary GFF only in the vertical direction. More precisely, (recall Remark REF ), the projection of \(h^w\)  onto \(H^L({S})\)  has the same law as the projection of a free boundary GFF \(h^f\)  on \({S}\)  onto \(H^L({S})\) . Furthermore, let \(Y_t\)  denote the average value of \(h^w\)  on \(\lbrace t\rbrace  \times (0,\pi )\) , that is, its projection on \(H^R({S})\) . Then the law of \(Y_t\)  is given by that of \(B_{2t} + (\gamma - \tfrac{2}{\gamma })t\)  where \(B\)  is a two-sided Brownian motion with \(B_0 = 0\)  and conditioned so that \(B_{2t} + (\gamma - \tfrac{2}{\gamma })t > 0\)  for all \(t > 0\) . If \(U_t\)  denotes the projection of \(h^f\)  onto \(H^R({S})\)  then the law of \(U_t\)  is equal to that of \(\widetilde{B}_{2t}\)  where \(\widetilde{B}\)  is a two-sided Brownian motion with \(\widetilde{B}_0 = 0\) . Consequently, it follows that if \(\mathfrak {h}^f\)  denotes the harmonic part of \(h^f\)  then for each \(Q \in \mathcal {Q}\) ,
\(\mathbf {E}\!\left[e^{\gamma p \sup _{z \in Q} |\mathfrak {h}(z)|} \right] \asymp \mathbf {E}\!\left[e^{\gamma p \sup _{z \in Q} |\mathfrak {h}^f(z)|} \right]\) 

where the implicit constant is independent of \(Q\) . For each \(k\)  we let \(\mathcal {Q}_k = \lbrace Q \in \mathcal {Q}: 2^{-k} \le \mathrm {dist}(Q,\partial {S}) < 2^{-k+1}\rbrace \) . Fix \(\delta > 0\) . By decreasing the value of \(\widehat{\delta } > 0\)  if necessary [2]} together with (REF ) implies for \(Q \in \mathcal {Q}_k\)  that
\(\mathbf {E}\!\left[e^{\gamma p \sup _{z \in Q} |\mathfrak {h}(z)|} \right] = O(2^{k(1+\delta ) \gamma ^2 p^2}),\) 

where the implicit constant depends only on \(\delta \) .
Next, note that \(|\mathcal {Q}_k| \asymp 2^k\)  with the implicit constant depending only on \(\delta \) . Moreover, since \(\mathrm {dist}(Q,\partial {S}) \asymp \mathrm {diam}(Q)\)  (with constants depending only on \(\delta \) ), we have for all \(Q \in \mathcal {Q}_k\)  that
\(\mathrm {dist}(Q,\partial {S})^{p\gamma ^2/2} |Q|^p \asymp 2^{-kp(2 + \gamma ^2/2)}\) 

where the implicit constants depend only on \(\delta \) . Combining this with (REF ) and (REF ) it follows that
\(\mathbf {E}\!\left[ \mathbf {E}\!\left[ \mu _{h^w}(A_0) \, | \, \mathfrak {h}\right]^p \right] \lesssim \sum _{k=0}^\infty O\!\left(2^{k((1+\delta ) \gamma ^2 p^2 -p(2+\gamma ^2/2)+1)}\right).\) 

Thus if \((1+\delta ) \gamma ^2 p^2 -p(2+\gamma ^2/2)+1 < 0\)  we have that the sum converges and the result follows. Note that the polynomial \(q(p) = \gamma ^2 p^2 -p(2+\gamma ^2/2)+1\)  attains its minimum \(-\tfrac{(4-\gamma ^2)^2}{16\gamma ^2} < 0\)  at the value \(p = \tfrac{4+\gamma ^2}{4\gamma ^2}\) . Consequently, for \(\delta > 0\)  very small there exists \(p_0 > 0\)  so that the polynomial \(q_\delta (p) = (1+\delta )\gamma ^2p^2 - p(2+\gamma ^2/2)+1\)  satisfies \(q_\delta (p_0) < 0\)  and hence such that (REF ) is bounded for \(p = p_0\) . By Lyapunov's inequality, the result holds for all \(0< p < p_0\)  as well.
We now turn to the bound on \(\mu _{h^w}(A_k)\)  for \(k \ge 1\) . By Hölder's inequality, with \(p^{\prime },q^{\prime } > 1\)  satisfying \(\tfrac{1}{p^{\prime }} + \tfrac{1}{q^{\prime }} = 1\) , we have that
\(\mathbf {E}\!\left[ \mathbf {E}[\mu _{h^w}(A_k) \,|\,\mathfrak {h}]^p \right] \le \mathbf {E}\!\left[ e^{p^{\prime } p \gamma Y_k} \right]^{\frac{1}{p^{\prime }}} \mathbf {E}\!\left[ \mathbf {E}[\mu _{h^w - Y_k}(A_k) \,|\,\mathfrak {h}]^{q^{\prime }p} \right]^{\frac{1}{q^{\prime }}}.\) 
Let \(\sigma = \inf \lbrace t \ge 0 : Y_t = 1\rbrace \) . Then we note that \(Y_{t+\sigma }\)  has the same law as \(Z_t = B_{2t} + (\gamma -2/\gamma ) t\)  conditioned on the positive probability event \(F\)  that \(Z_t > 0\)  for all \(t \ge 0\)  where \(B\)  is a standard Brownian motion with \(B_0 = 1\) . Let \({F}_t = \sigma (Y_s : s \le t)\) . Thus for \(b > 0\)  we have that
\(\mathbf {E}[ e^{b Y_t} ]&\le e^b + \mathbf {E}[ \exp (b Y_t) \mathbf {1}_{t \ge \sigma }]= e^b + \mathbf {E}[ \mathbf {E}[ \exp (b Y_t) \,|\,{F}_\sigma ] \mathbf {1}_{t \ge \sigma }].\) 

We moreover have on \(\lbrace t \ge \sigma \rbrace \)  that
\(\mathbf {E}[ \exp (b Y_t) \,|\,{F}_\sigma ]&= \mathbf {E}[ \exp (b Z_{t-\sigma }) \,|\,F]\le \frac{\mathbf {E}[ \exp (b Z_{t-\sigma })]}{\mathbf {P}[F]}\\&= \frac{\exp ( (b(\gamma -2/\gamma ) + b^2)(t-\sigma ))}{\mathbf {P}[F]}\\&\le \frac{\exp ( (b(\gamma -2/\gamma ) + b^2)t)}{\mathbf {P}[F]}.\) 

Inserting this into the above gives that
\( \mathbf {E}[ \exp (b Y_t)] \le e^b + \frac{1}{\mathbf {P}[F]} e^{(b(\gamma -2/\gamma ) + b^2)t}.\) 
Applying this for \(b = p p^{\prime } \gamma \)  and \(t =k\)  gives us that
\(\mathbf {E}\!\left[ e^{p p^{\prime } \gamma Y_k} \right] \lesssim e^{k((p p^{\prime })^2 \gamma ^2 + p p^{\prime } \gamma ^2 - 2p p^{\prime })}.\) 

The same argument works to bound the exponential moments of \((Y_{k+t}-Y_k)_{t \in [0,1]}\) . Hence,
\(\mathbf {E}\!\left[ \mathbf {E}[ \mu _{h^w - Y_k}(A_k) \,|\,\mathfrak {h}]^{q^{\prime }p} \right] \lesssim \mathbf {E}\!\left[ \mathbf {E}[\mu _{h^w}(A_0) \,|\,\mathfrak {h}]^{q^{\prime }p} \right] < \infty ,\) 

whenever \(q^{\prime }p\)  is sufficiently small. Hence, the second part of the lemma follows.
Finally, we bound the moments of the rectangles \(A_{-k}\)  for \(k \ge 1\) . We have that
\(\mathbf {E}\!\left[ \mathbf {E}[ \mu _{h^w}(A_{-k}) \,|\,\mathfrak {h}]^p \right]&= \mathbf {E}\!\left[ e^{\gamma Y_{1-k}}\mathbf {E}[ \mu _{h^w - Y_{1-k}}(A_{-k}) \,|\,\mathfrak {h}]^p \right]\\&\le \mathbf {E}[ e^{\gamma p p^{\prime } Y_{1-k}}]^{1/p^{\prime }} \mathbf {E}[ \mathbf {E}[ \mu _{h^w - Y_{1-k}}(A_{-k}) \,|\,\mathfrak {h}]^{p q^{\prime }}]^{1/q^{\prime }}\) 

The second term on the right-hand side is bounded by a constant times \(\mathbf {E}[ \mathbf {E}[\mu _{h^w}(A_0) \,|\,\mathfrak {h}]^{p q^{\prime }}]^{1/q^{\prime }}\)  (since the average process of the latter is a Brownian motion with positive drift, conditioned to stay positive, instead of an average process given by a Brownian motion with positive drift), which is finite for \(p > 0\)  small, and the former term is
\(\mathbf {E}[ e^{p p^{\prime } \gamma Y_{1-k}} ] \lesssim e^{(1-k) p p^{\prime } (\gamma ^2 - 2) + (p p^{\prime })^2 \gamma ^2 (k-1)} \lesssim e^{-k p p^{\prime } (\gamma ^2 - 2) + (p p^{\prime })^2 \gamma ^2 k}.\) 

where the implicit constant can be taken independent of \(k\) . Note that the exponent is negative for all \(\gamma \in (\sqrt{2},2)\)  and \(p, \varepsilon > 0\)  sufficiently small.
